Efficient Exact Sampling for the Ising Model at All Temperatures

The Ising model is often referred to as the most studied model of statistical physics. It describes the behavior of ferromagnetic material at different temperatures. It is an interesting model also for mathematicians, because although the Boltzmann distribution is continuous in the temperature parameter, the behavior of the usual single-spin dynamics to sample from this measure varies extremely. Namely, there is a critical temperature where we get rapid mixing above and very slow mixing below this temperature. Here, we present a technique that makes exact sampling of the Ising model on the square lattice at all temperatures possible in polynomial time. For this we use the results of the work that was done in the last 40 years together with the recent and seminal paper of Lubetzky and Sly that shows rapid mixing at the critical temperature. To conclude our result we need essentially the notion of duality of graphs and the Ising model on a slightly different graph.